Here are the things I don't know how to do: 1) Define a Laurent series by giving an expression for its n-th coefficient. Well, since you brought up metafont, I may as well suggest a sort of simple metafont solution. Linearity . .A. The ring of formal Laurent series over a commutative ring A A in an indeterminate x x consists of Laurent series ∑ n ∈ ℤ f n z n \sum_{n \in \mathbb{Z}} f_n z^n, with f n ∈ A f_n \in A but where all but finitely many f n f_n for n < 0 n \lt 0 vanish. Aparicio Monforte A(1), Kauers M(2). Tags: Discrete Valuation, Formal Laurent Series, Valuation Ring. An example from category 1 above is the field of Laurent series with a finite number of negative-power terms. Expo Math. This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. 2.2. Next Post Definition and properties of matrices with a single nonzero entry. On formal Laurent series On formal Laurent series Gan, Xiao-Xiong; Bugajewski, Dariusz 2011-09-01 00:00:00 Several kinds of formal Laurent series have been introduced with some restrictions so far. Continue Reading. It gets pretty messy so I was hoping that sage could keep track of the details for me. Several kinds of formal Laurent series have been introduced with some restrictions so far. In the multivariate case, it is not always possible to write a given f (x) ∈ K C 〚 x 〛 in the form f (x) = x e h (x) for some h (x) ∈ K C 〚 x 〛, as already illustrated in the introduction. Then M is a 1×1-matrix whose single entry is the smallest nonzero exponent appearing in g ( x ). The Field of Formal Laurent Series. Let Fp[T] be the polynomial ring with coeﬃcients in Fp and Fp(T) its fraction ﬁeld. Formal Laurent series in several variables. Add a comment | 1. On the other hand the second series in (0.1) is a regular power series, and hence setting R 2 = (limsup n!1 jc nj1=n) 1; the second series is convergent for jz aj1=R 1. We now introduce this ﬁeld. Consider a power series f(x) ∈ 〚x −1 〛 with negative exponents (i.e., C is the negative halfline) and let g(x) ∈ ((x))∖{0} be a usual formal Laurent series. 1. Author information: (1)Université Lille 1 Sciences et Technologies, 59655 Villeneuve d'Ascq, Cedex, France. Several kinds of formal Laurent series have been introduced with some restrictions so far. Defining formal Laurent series via K ((x)) ≔ ⋃ e ∈ Z x e K 〚 x 〛 therefore already leads to a field. I would like to compute some integrals of products of Laurent series, with the goal of getting an explicit expression for the n-th coefficient of the result. 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